Optimal. Leaf size=157 \[ -\frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5864, 5654, 5718, 5658, 3308, 2180, 2205, 2204} \[ -\frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5654
Rule 5658
Rule 5718
Rule 5864
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{4 d}\\ &=-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{8 d}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{4 d}+\frac {(3 b) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{4 d}\\ &=-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.76, size = 290, normalized size = 1.85 \[ \frac {\sqrt {\pi } \sqrt {b} (2 a-3 b) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {\pi } \sqrt {b} (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-12 b \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \sqrt {a+b \cosh ^{-1}(c+d x)}+8 b (c+d x) \cosh ^{-1}(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+4 a e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}}}\right )}{8 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________